Internat. J. Math. & Math. Sci.
VOL. 18 NO. 4 (1995) 789-798
789
COMPUTATION OF GREEN’S MATRICES FOR BOUNDARY VALUE PROBLEMS
ASSOCIATED WITH A PAIR OF MIXED LINEAR REGULAR ORDINARY
DIFFERENTIAL OPERATORS
T. GNANA BHASKAR and M. VENKATESULU
Department of Mathematics
Sri Sathya Sai Institute of Higher Learning
Prasanthinilayam, 515 134 INDIA
(Received July 29, 1991 and in revised form January 15, 1992)
ABSTRACT. An algorithm for the computation of Green’s matrices for boundary value problems
associated with a pair of mixed linear regular ordinary differential operators is presented and two
examples from the studies of acoustic waveguides in ocean and transverse vibrations in nonhomogeneous strings are discussed.
KEY WORDS AND PHRASES: Nonexplicitly mixed, matchingly mixed, boundary value
problem, Green’s matrices.
1991 AMS SUBJECT CLASSIFICATION CODES: 34XX, 34BXX, 34B27
INTRODUCTION
Recently, a new class of problems of the type where different differential operators are defined
over two adjacent intervals, involving certain mixed (interface) conditions are studied in [1,2,3,4].
1.
These problems involve a pair of differential operators of the type 1ul
-
ko0
PeDu
ul, defined
on the interval J-[a,b] and x2u2Q,D’u2 u2 defined on the adjacent interval J2-[b,c],
k-0
< a < b < +% where :k is an unknown constant (eigenvalue) and the functions u and u-z are
required to satisfy certain mixed conditions at the interface x b. In most of the cases, the complete
set of physical conditions on the system gives rise to selfadjoint eigenvalue problems associated
with the pair 0q,%). Based on the interface conditions these problems can be classified into three
types, namely (i) where the values of u and u-z are not explicitly related to each other at x b, (ii)
where u and u-z are required to satisfy the continuity conditions at x b, and (iii) where u and
satisfy certain matching conditions at x b.
The methods presented in [4] for the construction of Green’s matrices for the boundary value
problems (BVPs) associated with 0:a, x2) are theoretical in nature and involve lengthy calculations.
Here, in this paper we present (i) simpler algorithms for the computation of Green’s matrices for
the BVPs associated with 0:,), and (ii) construct the Green’s matrices for the problems found in
some physical situations.
Before indicating the division of the work into sections, we introduce a few notations and make
some assumptions. For any compact interval J of R and for a nonnegative integer k, let C(J) denote
the space of all k-times continuously differentiable complex valued functions defined on J. For a
function u, let u ) denote the j derivative of u, if it exists. For a compact interval J of R and for a
positive continuous (weight) function r(x) defined on J, let L(J,r) denote the Hilbert space of all
T. G. BHASKAR AND M. VENKATESULU
790
Lebesgue measurable complex valued functions u defined on J such that r(x)[u(x)[ is integrable
overJ. The inner product in L(J,r) is given by (u, v)-fu(x)v-r(x)dx, u, v L(J,r), where v(x’-’denotes the complex conjugate of v(x), and the norm is given by
I111- Cx)luCx)l d
. eL2(J,r)
Let H(J,r) denote those functions inACff) such that both u and u ) are inL ff,r). t C denotes
the k-dimensional complex space whose elemenm we take to be column vectors. For k x matrix
A with complex entries, A" denote the x k matrix which is the conjugate transpose ofA. t A
denote the inverse of a square matrix A, if it exists. If V and V are vector spaces (over the same
field), then V x V denotes the cartesian product of V and V taken in that order. For an operator T,
D(T),R(T),N(T), (T) denote the domain, range, null space and the dimension of the null space of
T, respectively. t X- L ff,r) x L (J r) be the caesian product Hilbert space equipped with the
inner product (-) and the no "11 given by
-
<{u,. }, {v,, v}>
<,, v,> + <u.v, {,, u}. {,, v} ex,
and
{.,,.}II (II.,II
+
II.II’)’=,
{.,,}
Let H H’(Jl, rl) H"(J2,r2) be the cartesian product Banach space.
.
.
ASSUMPTION 1. LetJl-[a,b]andJ2-[b,c],-<a <b
0 Q’D’
on J1; Q
<c < +.
be two formal differential expressions, where P
C(J2), k O, 1
m,Q,,,(x) 0 on J2; and r,(x)
real valued functions. We also assume n
Letx,-,,’- ._ PD
C(J), k O, 1
and
n, P,(x) ,, 0
C(J,) and r:,(x) C(J) are positive
m.
ASSUMPTION 2. LetA andB hem n andre tn matrices with complex entries, respectively
such that the range ofA range of B, and hence, rank of A rank ofB -m.
In Section 1, we shall collect together a few definitions and results, from our earlier papers,
which we require here. In Section 2, we shall present a lemma regarding the form of solutions of
a type of initial value problems (IVPs) associated with the pair (x,,x:,), in terms of Green’s matrices.
In Section 3, we shall present an algorithm for the computation of Green’s matrices for the BVPs
associated with the pair (x,,x:0- In Section 4, we shall construct the Green’s matrices for problems
encountered in the studies of acoustic wave guides in ocean and transverse vibrations in nonhomogeneous strings.
2.
PRELIMINARIES
Letfbe a complex valued function defined onJ. Let f -f/J,
(,,)u -/"
1,2. LetJ -J LIJ2. Consider
(2.)
and the corresponding homogeneous equation
(,,’r,2)u 0.
(2.2)
DEFINITION I. We call a complex valued function u(x) defined on the interval J, a solution
(nonexplicitly mixed) of (2.1) if
(i) the functions u/J u AC"(J) and u/J u AC"(J)
791
GREEN’S MATRICES FOR BOUNDARY VALUE PROBLEMS
(ii)
u and u satisfy the equations xul f, for x J1 a.e., and :2u2 f2, for x J2 a.e., respec-
DEFINITION 2. We call a complex valued function u(x) defined on the interval J, a continuous
solution of (2.1) if
(i) u is a solution of (2.1) in the sense of Definition 1, and
(ii)
the functions ui and u satisfy the continuity conditions at the interface point x b, namely,
u)(b -)= u)(b +),
j
O,
m
1.
DEFINITION 3. We call a complex valued function u(x) defined on the interval J, a matching
solution of (2.1) if
(i) u is a solution of (2.1) in the sense of Definition 1, and
(ii) the functions u and u satisfy certain matching conditions at the interface point x-b,
namely, A a (b) B t2(b), where
t2 t(b) column(ul(b), u)(b)
u" -X)(b)),
and
z(b
,u2t’-(b ))
column(uz(b ),ut2X)(b
REMARK 1. All the above definitions can be carried over to equation (2.2) also.
Below, we recall a few definitions from [6], in the form, required here. Let x
DEFINITION 5. The nonexplicitly mixed operator N0:) is defined by
D(N(’r,))-- {{ut, u,} H/Bi’V({u,u2})- O,
n
+m},
o(a,u’(a)+ou)(b))+ o(?,iu’(b)+,iu’(c))
i- 1
() {u,
u
(u,/,
where
B({u,u2} )-
-1
m-1
n +m
are the linearly independent nonexplicitly mixed bounda values.
DEFITION 6. The continuously mixed operator C() is defined by
O(C(x))-{{u,u:} eH/Bf({u,})-O,
c(x) {ut,} {xu,x}) where
n,
i-1
u)(b)-u2)(b), i-1,...,n},
N-1
,{u,u- .0 ,u><a+O,u><)+6,u’<)
i- L...,n
are the linearly independent continuously mixed bounda values.
DEFION 7.
e matchingly mixed operator M(x) is defined by
D(M(x))- {{u,u2} eH/Bf({u,u})-O,
1
n +m,
Aa(b)-B(b),}
M() { ul, u}
where
BiC{u,u:,})
N-1
,,oCa,u’)(a) + bou’)(c)) +
,
a(b
are the linearly independent matchingly mixed boundary values.
i- 1
n,
792
T.G. BHASKAR AND M. VENKATESULU
REMARK 2. For the sake of brevity, we shall study only the operators N(r) and M(r) and the
results for the operator C(r) follow by takingA -B -I (the n n identity matrix) in the results for
M(r).
ASSUMPTION 3. For the matchingly mixed case we assume that n m.
2. LEMMA REGARDING THE IVPs ASSOCIATED WITH O:,r)
We consider a particular type of initial value problem associated with (rt, r2) for nonexplicitly
mixed and matchingly mixed operators and give a result about the form of the solution of the IVPs,
in terms of Green’s matrices.
(I) Nonexplicitly Mixed Initial Value Problems
Let ul
u,,l
and u12
u,a be functions in H(J,r) and Hm(J,r2) which form bases for
the solution spaces of
rut-0 and r2u2-0, respectively. Then, the pairs {ul,0},
{u,0} {u,,,0}, {0,ul2} {0,u,,a} (all of which belong to H) form basis for the solution space
ofN(r){u,u2} -0 (for the explicit form of the basis see [3]).
Define N(z) to be the operator in H such that
NCr) {{u,u2} H/u()(a)=O, j -0
n
REMARK 3. We note that the Wronskian of ul
1,
u)Cb)-O, j -0
u,,, namely, W(Ull
1},
m
Unl)(S
0 for all
J1, and the Wronskian of u:,l, u,,,:, namely W(u2
urn2) (s) 0 for all s J2. And, we denote
the
determinant
obtained
by W(u,
un)(s)
by replacing the ’’ column in the corresponding
Wronskian by (0, 0
1) C’,i 1 n. Similarly, we define W/(u12, u,,,2)(s).
(II) Matchingly mixed initial value problems
Let the set of pairs {u.,ul}
{u,l,u,,2} be a basis for the solution space ofMo(r) {ut, u} 0,
s
where
Mo(r) {{u,u} H/Aa(b)-Bu2(b)}
M0() {u,u_} {u,}.
Also, define M(r) to be the operator in H such that
M(r) { {u, u} H/u)(a O, j O, ...,n 1,aa(b Bud(b)},
M(r) {u, u2} {ru,r.zu}.
The lemma below follows from the variation of parameters formula.
LEMMA. (I) The solution {u,u-z} ofN(r){u,u2} {f,j} is of the form u(x)- {u(x),u(x)}
;
where
G(x,s)f(s)r(s)ds
xJ
G(x,s)f2(s)r(s)ds
x J2
GREEN’S MATRICES FOR BOUNDARY VALUE PROBLEMS
o,
w.(u
u.,)(s)
w, Cu,
u.)(s)
u,(x)
,u,,,2)(s)
Gzz(X’S)’i-lQ,,,(s)WCulz
u,(x)
793
a <s <x <b,
b <s <x <c,
Also, we define
(2.3)
and call G ’v as the Green’s matrix for N"0:).
II) The solution {ul, uz} ofM(z){u,u2} {f,f2} is of the form u(x)- {u(x),u2(x)}
/ Gff(x,s)f(s)rx(s)ds
x
a(x,s)(s)rl(s) / a,’(x,s)(s)r,(s)
where
W,(u
,u.3(s)
c,(x,s)-,..p.(s)W(u
a <s <x <b,
a<s<b, b<x<c,
u.O(s)
and
W,(u.
::
u.)(s)
u,(x)
b
<$
<x <c
Also, we define
Gu
G2ff
0
(2.4)
G
and call G u as the Green’s matrix for M(r,).
COMPUTATIONAL ALGORITHM FOR THE GREEN’S MATRICES FOR
OPERATORS ASSOCIATED WITH
In this section, proceeding along the lines of [5], we present an algorithm for the computation
of Green’s matrices for operators associated with
(I) Nonexplicitly mixed operator: Consider {fx,j} X.
3.
Let u(x) {ux(x),uz(x)} (N0:))-x (f,J}. Then (see [4]), u(x) {Ul(X), U2(/)}
f. G(x,s)f(s)r(s)ds+ f G(x,s)f(s)r,(s)ds,
f G(x’s)fl(s)rl(s)ds+ f, G(x’s)f(s)r2(s)ds’
we denote
(3.1)
x J.
T. G. BHASKAR AND M. VENKATESULU
794
-
"
Let v(x)-{v(x),v2(x)}
(N(x)) {A(x),f(x)}. By Theorem 4 [1], we have rl(N(r)) n + m. Since, {ui vi, u2- v2} belongs
to the solution space of x{u,u} -O, there exists scalars c
cn such that u(x)- {u(x),u2(x)},
and we call G
the Green’s matrix for the operator N(x.).
E cn .iu2(x) + v(x)
qu,(x) + v(x),
-1
(3.3)
i-I
Applying the boundary value on (3.3), we have B({u,u}) 0
1,2, ...,n + m. That is,
B:({ ciuil(X", c"/iui(x’})"-B:({v"v2}’
(3.4)
"-t
.t
But,
where
B:i- ,.i(c:tu ")(a) + i,u)(b))
1
n
and
t.
?itu)(a + 6itu)(b
1,
m
1,
n +m
Relation (3.4) can now be written as,
i-I
,
c.,B:, + c. /.,B, -Bf({vx, v2}),
I, ...,n + m.
It can be verified that the coefficient matrix of the (n + m) (n + m) linear system (3.5) in (n + m)
unknowns, is nonsingular. Now, by the choice of {vl, v2}, we have
-1
-1
where
and
m-1
-,C)
Clearly,
Y_, ,E
N H"(J,r)
(u.
u.:O()
and NH’(J,r).
tokut tc
Rewriting (3.5), we have, for
(3.6)
GREENWS MATRICES FOR BOUNDARY VALUE PROBLEMS
LetB-(B/,),i- 1
n
795
andB--(B),j-O,...,m l- 1,...,n +m. LetB -[B,B]. It can beshown
0. Consider the (n + in) (n + m) linear system, for
that B is a nonsingular matrix. That is, det B
l- 1,...,n +m,
i-l
-
B,’,{z,,,z,} + B{z,,/,),z,./,} --{,},
We have by Cramer’s rule, {z,(s),ziz(s)}
-
{B,,B},i
(3.7)
n and j-1,2,, m where B and
1,.
B are determinants obtained by replacing the ’ and j’ columns inB andB, by the column vectors
,+,), respectively. at is, each of z(s) and z(s) are linear
combinations off,(s) andre(s), respectively. Hence, {z(s),z(s)} H. en, we have by taking
the inner-product of both sides of (3.7) with {A,},
(t,...,,.,>) and (g
-1
-1
(by (3.6)), which implies that,
c ({zl(s),z(s) }, {ft(s),f2(s)}),
1
n + m.
(3.8)
Combining (3.3) and (3.8), and comparing with (5), we get,
G (x,s)a <x <s < b
Uil(X)Zil(S)
ui(xi(s ), a
<x <b
b <s <c
G21(x,s)- i-I ui(x)zi(s ), b <x <c
a <s <b
62(x,s )-
6(x,s)-
[ ilui(x)z(s)+G(x,s),
ui(x )zi(s
b <s <x <c
b <x <s < c
This completes the algorithm for the computation of Green’s matrix G N for the nonexplieitly mixed
operator N:).
REMARK 4. The algorithm for the computation of the Green’s matrix G u for the operator
M(x), runs along the similar lines, with n m.
PHYSICAL EXAMPLES
In this section, we shall use the computational algorithms developed in Section 3, to compute
the Green’s matrices for a matchingly mixed operator an/] a continuously mixed operator,
4.
encountered in the studies of acoustic waveguides in oceans and transverse vibrations in nonhomogeneous strings, respectively.
T. G. BHASKAR AND M. VENKATESULU
796
(I) Acoustic waveguides in oceans [6]:
Consider the ocean to be consisting of two homogeneous layers, with a rigid bottom and a
pressure release surface. Then, the propagation of acoustic waveguides in such an ocean is governed
by the following equations.
r,u,
u + K2 u
z)
Lug, 0 < x < d
<x <
+
together with the mixed boundary conditions given by,
o,
u (0)
u (clg
.
where p and P2 are constant densities of the two layers, K,K2 are constants which depend upon the
frequency constant o and the constant sound velocities cl, c2 of the two layers, respectively, is an
unknown constant, [0,d] and [dl, d2] denote the depth of the two layers and u and u2 stand for the
depth eigenfunctions. LetJ [O,d] and J2 [d,d2]. The matching conditions at the interfacex --dl
can be written in the matrix form Axa(d)-B2a2(dt), where ai(dO-column(ui(di),u.t,n(dO),
Ai
1/pi
for
1,2. Also, we have n
M(x) { {u, u2}
m
d
2. Define
H2(J, 1/p) x H2(J2,1/p:) /a fl l(d)
A fl(d,), u(O)
u’)(d2) 0},
M(r,)u {’u,xzuz}
After simple calculations along lines of the algorithm, we get the form of the Green’s matrix G u to
be of the form,
sinKx
(19.:,K cosK(d
KM
G- sinK:
d) cosK(cl
s
K sinK(cl
s
sinK(cl d))
[ KM (PcsK(-d0csK(d-x)-pKsinK(d-x)sinK(-d0)’
p’
cosK2(d2-s),
G -s,nKxx
0<x <all,
di <$ <d
sinK: cosK2(dz-x ),
0 <s <d,
d <x <d
0 x <s
d
0 <s <x
[osK(a-,)
G
J|cosK2(d_x)(PzK sinK,d, csK2(d,-s)+P:,K, cosK,
d sinK(s
-d,)), d, <s <x
We also note that
-M
s)
AtdCdt, s) -A2G2t(dt,
and
AJ(d,,s AG(a,s
REMARK 5. In the above, we have the compact and general form of the Green’s matrix of
the problem compared to the one given in [6].
GREEN’S MATRICES FOR BOUNDARY VALUE PROBLEMS
797
(II) Transverse vibrations in nonhomogeneous strings [7]:
Consider the string consisting of two portions of lengths dl and d- dl, and different uniform
.
densities p, P2 respectively, having tension T and stretched between the points x
modes of transverse vibrations of the above string are governed by,
.- c(-.))
and
0 andx
d2. The
o < x < d.
2.
(2)
:2u2-c2t-u )-Lu2, dl<x<d2,
together with the mixed boundary conditions given by,
u(0) u:Cd9 0. u(4) u(d)
where c
T/pi,
u)(d) u)(4)
1, 2. Here, the conditions at the interface point are the continuity conditions.
Proceeding along the lines of the algorithm, we get, after routine calculations, the Green’s
matrix G to be of the form,
,
,
,
0<x<dl, d<s<d
0<x<s<d
0<s<x<d
c.
We note that _,(dx, s)- t2C(dx, s) and similar relations are true of the components G and G
ACKNOWLEDGMENT
The authors dedicate the work to the chancellor of the institute Bhagavan sri sathya sai baba.
REFERENCES
[1] VENKATESULU, M. and BHASKAR, T. GNANA, Solutions of initial value problems
[2]
[3]
[4]
[5]
[6]
[7]
associated with a pair of mixed linear ordinary differential equations, J. Math. Anal_. Appl,
146 (2), 1991, 63-78.
VENKATESULU, M. and BHASKAR, T. QNANA, Fundamental systems and solutions of
nonhomogeneous equations associated with a pair of mixed linear ordinary differential
equations, J. Aust. Math, Soc, series A49, 1990.
VENKATESULU, M. and BHASKAR, T. GNANA, Selfadjoint boundary value problems
associated with a pair of mixed linear ordinary differential equations, J. Math. Anal. Appl.
144 (2), 1989.
VENKATESULU, M. and BHASKAR, T. GNANA, Green’s matrices for boundary value
problems associated with a pair of mixed linear regular ordinary differential operators, to
appear in Zeitschrift f. Anal.
LOCKER, JOHN, Functional analysis and two point differential operators, Pitman Research
Notes, 1986.
BOYLES, C. ALLAN, Acoustic Waveguid.es, Applications to O.ceani Sciences, Wiley, New
York, 1984.
GHOSH, P. K...The Mathematics of Waves and Vibrations, Macmillan, Delhi, India, 1975.